Sinn Fein president Gerry Adams said a resolution was possible, with talks recommencing today and expected to continue into next week.
Former US diplomat Richard Haass is expected back in Belfast today to chair another round of discussions and has been in contact with the parties over the festive break.
Separate talks with representatives of the five main parties are likely before full negotiations on a fresh set of proposals.
Attempts failed in the run-up to Christmas and Dr Haass and his vice-chairman Dr Meghan O'Sullivan returned home - but indicated they were prepared to come back to Belfast if they thought there was a realistic chance of agreement on at least some of the issues.
Mr Adams said: "There is a duty and responsibility on all the parties to these negotiations, despite the challenges, to find a way forward. With a fair wind the proposals under consideration can do this and I would appeal to everyone to overcome any difficulties which may remain."
It is believed a fifth draft will be presented to the parties tomorrow and they will have until New Year's Eve to come up with a settlement.
All sides believe there is a possibility they can agree on the way forward on parades and the past, but the question of flags is almost certain to remain unresolved for the foreseeable future.
Mr Adams, former West Belfast MP and now a member of the Irish Dail, added: "It is our view that progress was made and that agreement is possible when the talks recommence."
Dr Haass and Harvard professor Dr O'Sullivan were brought to Northern Ireland in July by the First Minister Peter Robinson and the Deputy First Minister Martin McGuinness with an aim of finding consensus on the displaying of national flags, the holding of parades and how to come to terms with the violence of the past.